I drove my wife and 9-year-old twins (they turned 9 during the trip!) from DC to Ft. Lauderdale this summer, with southbound stops in Myrtle Beach and Savannah, and a northbound stop in Charleston. My wife was actually the most apprehensive about it out of all of us, and her reaction after the fact was that it wasn't as bad as she thought it would be.
The kids mainly read books and played on Android tablets in the car. The "forced breaks" every 2-3 hours for supercharging worked out perfectly for us -- we always needed a break of some sort around that interval anyway. And with kids age 1 and 3, you might even need breaks more often than the superchargers.
If you have the time to drive, and your time in FL will be at least a week, I'd say go for it! Though, in the interest of full disclosure, our next trip to FL will be by plane, mainly because we won't have as much time or flexibility, and also because we're not planning on any interim destinations.
Depending on where in FL you're going, unless of course you're staying really close to a supercharger, be sure to plan your destination charging. We mainly used 120V outlets at our South Carolina and Florida destinations, which worked just fine to cover daily local driving and build up enough range to get to the next supercharger. (BTW, consider buying the NEMA 5-20 adapter if you don't already have it. We ended up not using it because everything we found was 15A, but many commercial buildings have 20A outlets outdoors.)
I-95 supercharging is generally super-easy, though there are a few locations with limited amenities. We couldn't find all that much to do while supercharging in Rocky Mount or Savannah, for example, so they were really just bathroom breaks. I can't comment on the superchargers north of Hamilton NJ (yet). I had one issue at Port St. Lucie where the car wouldn't charge; I called the supercharger support phone number after trying 2-3 stalls; the rep on the phone could tell me almost immediately that only 2 stalls were working at the time, and told me which ones. My other issue was that the adhesive on the magnet keeping my charge port closed failed during the southbound trip, but it didn't prevent the car from driving (I called roadside assistance to make sure it would be safe to continue), and the Dania Beach service center took great care of me (valet service and all) while I was in Ft. Lauderdale.
I found EVTripPlanner.com to be quite accurate in calculating the range needed between superchargers. I wrote myself a small cheat-sheet with all the expected numbers for each leg, and my actual rated-range usage always ended up somewhere between actual miles and EVTripPlanner's estimate.
6-8 hours of driving per day tells me you'll probably need ~3 overnight stops, again depending on where in FL you're going. You've likely considered this already, but try to make each overnight destination something at least moderately fun/interesting for the kids. Even if it's as minor as finding hotels along I-95 with indoor swimming pools.
In terms of breakdown concerns, well, as others have mentioned it's a possibility in any car. Part of my wife's concern I think stemmed from a childhood trip she took with her family, where their car broke down 3x en route, and they had to turn around before their final destination. I did have a flat tire kit in the car, but I didn't buy it just for the trip. Worst case for Tesla, there's roadside assistance / ranger service.